Considerações finais

A propriedade fractal dos depósitos IOCG de Carajás indica haver padrões sistemáticos na geometria dos depósitos que se conservam independente da escala espacial, mesmo considerando a complexidade inerente aos processos metalogenéticos. O conjunto de resultados obtidos nesta pesquisa permite inferir que padrões de escala microscópica de minerais de minério, bem como estruturas de escala local próximas de depósitos minerais, apresentam relações geométricas consistentes com as estruturas regionais às quais estão relacionadas, principalmente devido à invariância escalar que apresentam. Dessa forma, parece ser possível extrair informações úteis das escalas microscópica e local para inferir algumas das propriedades dos controles estruturais regionais. Além disso, controles estruturais inferidos a partir de feições das escala microscópica e local apresentam potencial para a previsão da favorabilidade mineral regional, uma vez que permitem individualizar estruturas mais favoráveis à mineralização e, portanto, mais prospectivas.

O conjunto de inferências deste trabalho sugere que a geometria fractal é uma ferramenta útil à prospecção mineral, e indicam que futuros trabalhos podem identificar aplicações para as técnicas aqui apresentadas. A abordagem aqui utilizada tem potencial particularmente interessante para greenfields, onde é comum a falta de dados e conhecimento sobre depósitos minerais e seus possíveis controles de mineralização.

REFERÊNCIAS

Berman M. 1977. Distance distributions associated with Poisson processes of geometric figures. Journal of Applied Probability, 14(1):195–199.

Blenkinsop T.G. and Sanderson D.J. 1999. Are gold deposits in the crust fractals? A study of gold mines in the Zimbabwe craton. In: McCaffrey K.J.W., Lonergan L., & Wilkinson J.J. (eds.) Fractures, Fluid Flow and Mineralization. Special Publications, 155, London, Geological Society, p. 141–151.

Carlson C.A. 1991. Spatial distribution of ore deposits. Geology, 19(2):111–114.

Carranza E.J.M. 2008. Geochemical Anomaly and Mineral Prospectivity Mapping in GIS. Amsterdam, Elsevier B.V., 366 p.

Carranza E.J.M. 2009. Controls on mineral deposit occurrence inferred from analysis of their spatial pattern and spatial association with geological features. Ore Geology Reviews, 35(3–4):383–400.

Carvalho E. de R. 2009. Caracterização Geológica e Gênese das Mineralizações de Óxido de Fe-Cu-Au e metais associados na Província Mineral de Carajás: Estudo de Caso do Depósito de Sossego: Universidade Estadual de Campinas, 141 p.

Cox S.F., Knackstedt M.A. and Braun J. 2001. Principles of structural control on permeability and fluid flow in hydrothermal systems. Reviews in Economic Geology, 14:1–24.

Davis G.H. and Reynolds S.J. 1996. Structural Geology of Rocks and Regions. Wiley, 800 p. Delvaux D. and Sperner B. 2003. New aspects of tectonic stress inversion with reference to

the TENSOR program. In: Nieuwland D.A. (ed.) New Insights into Structural Interpretation and Modelling, London, Geological Society, p. 75–100.

Domingos F.H.G. 2009. The structural setting of the Canaã dos Carajás region and Sossego- Sequeirinho deposits, Carajás Brazil [Doctor of Philosophy thesis]: University of Durham, 483 p.

Ford A. and Blenkinsop T.G. 2008. Evaluating geological complexity and complexity gradients as controls on copper mineralisation, Mt Isa Inlier. Australian Journal of Earth Sciences, 55:13–23.

Fry N. 1979. Random point distributions and strain measurement in rocks. Tectonophysics, 60(1–2):89–105.

Hansen V.L. 1990. Collection and preparation of thin sections of oriented samples. Journal of Geological Education, 38:294–297.

165

Heilbronner R. and Barrett S.D. 2014. Image Analysis in Earth Sciences. Berlin Heidelberg, Springer Verlag, 520 p.

Hippertt J. 1999. Are S-C structures, duplexes and conjugate shear zones different manifestations of the same scale-invariant phenomenon? Journal of Structural Geology, 21(8–9):975–984.

Hippertt J.F. and Massucatto A.J. 1998. Phyllonitization and development of kilometer-size extension gashes in a continental-scale strike-slip shear zone, north Goiás, central Brazil. Journal of Structural Geology, 20(4):433–445.

Hitzman M.W. 2000. Iron Oxide-Cu-Au Deposits: What, Where, When and Why. In: Porter T.M. (ed.) Hydrothermal Iron Oxide Copper-Gold & Related Deposits: A Global Perspective, Volume 1, Adelaide, PGC Publishing, p. 9–25.

Hodkiewicz P.F., Weinberg R.F., Gardoll S.J. and Groves D.I. 2005. Complexity gradients in the Yilgarn Craton: Fundamental controls on crustal-scale fluid flow and the formation of world-class orogenic-gold deposits. Australian Journal of Earth Sciences, 52(6):831– 841.

Holdsworth R.E. and Pinheiro R.V.L. 2000. The anatomy of shallow-crustal transpressional structures: Insights from the Archaean Carajas fault zone, Amazon, Brazil. Journal of Structural Geology, 22(8):1105–1123.

Jébrak M. 1997. Hydrothermal breccias in vein-type ore deposits: A review of mechanisms, morphology and size distribution. Ore Geology Reviews, 12(3):111–134.

Jensen E., Cembrano J., Faulkner D., Veloso E. and Arancibia G. 2011. Development of a self-similar strike-slip duplex system in the Atacama Fault system, Chile. Journal of Structural Geology, 33(11):1611–1626.

Jerram D.A., Cheadle M.J., Hunter R.H. and Elliott M.T. 1996. The spatial distribution of grains and crystals in rocks. Contributions to Mineralogy and Petrology, 125(1):60–74. Johnston J.D. and McCaffrey K.J.W. 1996. Fractal geometries of vein systems and the

variation of scaling relationships with mechanism. Journal of Structural Geology, 18(2– 3):349–358.

Kretz R. 1969. On the spatial distribution of crystals in rocks. Lithos, 2(1):39–65.

Kruhl J.H. 2013. Fractal-geometry techniques in the quantification of complex rock structures: A special view on scaling regimes, inhomogeneity and anisotropy. Journal of Structural Geology, 46:2–21.

Leite E.P. and Souza Filho C.R. de. 2009. Artificial neural networks applied to mineral potential mapping for copper-gold mineralizations in the Carajás Mineral Province,

Brazil. Geophysical Prospecting, 57(6):1049–1065.

Mandelbrot B.B. 1983. The Fractal Geometry of Nature (Updated and Augmented Edition). New York, Freeman, 495 p.

Marshak S. and Mitra G. 1988. Basic Methods of Structural Geology. Englewood Cliffs, NJ, Prentice-Hall, 446 p.

Monteiro L.V.S., Xavier R.P., Carvalho E.R. de, Hitzman M.W., Johnson C.A., Souza Filho C.R. de and Torresi I. 2008a. Spatial and temporal zoning of hydrothermal alteration and mineralization in the Sossego iron oxide–copper–gold deposit, Carajás Mineral Province, Brazil: paragenesis and stable isotope constraints. Mineralium Deposita, 43(2):129–159.

Monteiro L.V.S., Xavier R.P., Hitzman M.W., Juliani C., Souza Filho C.R. de and Carvalho E.R. de. 2008b. Mineral chemistry of ore and hydrothermal alteration at the Sossego iron oxide-copper-gold deposit, Carajás Mineral Province, Brazil. Ore Geology Reviews, 34(3):317–336.

Morais R.P.S. de and Alkmim F.F. de. 2005. O controle litoestrutural da mineralização de cobre do depósito Sequeirinho, Canaã dos Carajás, PA. In: 1o Simpósio Brasileiro de Metalogenia, Gramado, Sociedade Brasileira de Geologia, p. CD-ROM.

Moreto C.P.N., Monteiro L.V.S., Xavier R.P., Creaser R.A., DuFrane S.A., Melo G.H.C., Silva M.A.D. da, Tassinari C.C.G. and Sato K. 2015a. Timing of multiple hydrothermal events in the iron oxide–copper–gold deposits of the Southern Copper Belt, Carajás Province, Brazil. Mineralium Deposita, 50(5):517–546.

Moreto C.P.N., Monteiro L.V.S., Xavier R.P., Creaser R.A., DuFrane S.A., Tassinari C.C.G., Sato K., Kemp A.I.S. and Amaral W.S. 2015b. Neoarchean and Paleoproterozoic Iron Oxide-Copper-Gold events at the Sossego Deposit, Carajás Province, Brazil: Re-Os and U-Pb geochronological evidence. Economic Geology, 110:809–835.

Nykänen V. and Ojala V.J. 2007. Spatial analysis techniques as successful mineral-potential mapping tools for orogenic gold deposits in the northern Fennoscandian shield, Finland. Natural Resources Research, 16(2):85–92.

Oliver N.H.S. 1996. Review and classification of structural controls on fluid flow during regional metamorphism. Journal of Metamorphic Geology, 14:477–492.

Ord A., Munro M. and Hobbs B. 2016. Hydrothermal mineralising systems as chemical reactors: Wavelet analysis, multifractals and correlations. Ore Geology Reviews, 79:155–179.

167

transcorrentes Carajás e Cinzento, Cinturão Itacaiúnas, na borda leste do Cráton Amazônico, Pará. Revista Brasileira de Geociências, 30(4):597–606.

Raines G.L. 2008. Are fractal dimensions of the spatial distribution of mineral deposits meaningful? Natural Resources Research, 17(2):87–97.

Sibson R.H. 1994. Crustal stress, faulting and fluid flow. Geological Society, London, Special Publications, 78(1):69–84.

Sibson R.H. 1996. Structural permeability of fluid-driven fault-fracture meshes. Journal of Structural Geology, 18(8):1031–1042.

Turcotte D.L. 1989. Fractals in geology and geophysics. Pure and Applied Geophysics, 131(1/2):171–196.

Turcotte D.L. and Huang J. 1995. Fractal Distributions in Geology, Scale Invariance, and Deterministic Chaos. In: Barton C.C. & La Pointe P.R. (eds.) Fractals in the Earth Sciences, Boston, MA, Springer US, p. 1–40.

Vasquez M.L., Rosa-Costa L.T. da, Silva C.M.G. da and Klein E.L. 2008. Compartimentação Geotectônica. In: Vasquez M.L. & Rosa-Costa L.T. da (eds.) Geologia e Recursos Minerais do Estado do Pará: Sistema de Informações Geográficas - SIG: texto explicativo dos mapas Geológico e Tectônico e de Recursos Minerais do Estado do Pará, Belém, CPRM, p. 39–112.

Vearncombe J.R. and Vearncombe S. 1999. The spatial distribution of mineralization: Applications of Fry analysis. Economic Geology, 94(4):475–486.

Wang Z., Cheng Q., Cao L., Xia Q. and Chen Z. 2007. Fractal modelling of the microstructure property of quartz mylonite during deformation process. Mathematical Geology, 39(1):53–68.

Weinberg R.F., Hodkiewicz P.F. and Groves D.I. 2004. What controls gold distribution in Archean terranes? Geology, 32(7):545–548.

Williams P.J., Barton M.D., Johnson D.A., Fontboté L., Haller A. de, Mark G., Oliver N.H.S. and Marschik R. 2005. Iron oxide copper-gold deposits: Geology, space-time distribution, and possible modes of origin. Economic Geology 100th Anniversary Volume,:371–405.

Xavier R.P., Monteiro L.V.S., Moreto C.P.N., Pestilho A.L.S., Melo G.H.C. de, Silva M.A.D. da, Aires B., Ribeiro C. and Silva F.H.F. e. 2012. The Iron Oxide Copper-Gold Systems of the Carajás Mineral Province, Brazil. In: Hedenquist J.W., Harris M., & Camus F. (eds.) SEG Special Publication, 16, Society of Economic Geologists Inc., p. 433–454.

APÊNDICE A – Material suplementar do CAPÍTULO 2

Supplementary Material

Here we expand and complement key concepts on Fry and fractal analysis used in the manuscript.

1. Randomness of point patterns and suitability of Fry analysis

For point objects, a completely random distribution is the result of a Poisson process (Fry, 1979; Boots and Getis, 1988), in which the point pattern follows two basic conditions: (i) uniformity – any location in the studied spatial pattern has equal chance of a point occurrence; and (ii) independence – the occurrence of a point at a certain location in the spatial pattern does not influence the chance of occurrence of another point at another location. The resulting distribution is known as complete spatial randomness (CSR) (Diggle, 1983). The importance of a distribution with CSR is that it serves as an ideal reference for the definition of two basic patterns of point distributions (Fig. 1, Boots and Getis, 1988): (i) a clustered pattern, in which points display a significantly higher grouping than is expected in CSR (Fig. 1b); and (ii) a regular pattern, in which points show a spread that is significantly higher than is expected in CSR (Fig, 1c). A clustered pattern suggests objects resulting from an interplay of processes that involve ‘concentration’ of groups of such objects into certain locations; whereas a regular pattern suggests objects resulting from an interplay of processes that involve ‘circulation’ of such objects toward certain locations (Carranza, 2009).

Clustered and regular patterns can be generated by the lack of at least one of the basic conditions for CSR, i.e., non-uniformity or dependence. As a consequence, such patterns can contain some randomness, meaning they are not completely random. Fry analysis is suitable for these types of patterns.

169

Fig. 1: Fundamental types of point pattern distributions (from Boots and Getis, 1988): (a) complete

spatial randomness (CSR); (b) clustered; (c) regular.

2. Dimension concepts and the origin of the fractal dimension

Dimension is a concept usually defined as "the number of parameters needed to uniquely identify all points of space". This dimension concept can be extended to higher dimensional spaces, and is known as Euclidean dimension. Although useful, the definition of the Euclidean dimension of certain objects is unsatisfactory in certain situations, so that mathematicians of the transition between the 19th and 20th centuries proposed an alternative definition, known as the topological or Lebesgue dimension (DT). In this concept, the

dimension of a space is the largest of its local dimensions, where the local dimension is defined as the sum of a unit and the Euclidean dimension of the smallest object capable of dividing the original space into two parts. For example: a point is the object with the smallest Euclidean dimension (0), which is able to separate a line into two parts. So the DT for a line is

1 (i.e., 1 + 0). A line is the object with the smallest Euclidean dimension (1) which is capable of separating a plan (or area) in two parts. So, the DT for a plane is 2 (i.e., 1 + 1). Although

wider than the Euclidean concept, the concept of topological dimension also shows limitations, illustrated by features such as the Koch and Peano curves (Fig. 2).

Fig. 2: First six iterations for the construction of a Hilbert curve, a variation of the Peano curve

(modified from “Hilbert curve”. Licensed under CC BY-SA 3.0 via Wikipedia). In the figure scale, the sixth iteration already reveals a curve with indistinguishable contours. These curves are known as space-filling curves, precisely because as the iterations advance, their length tends to fill up the whole plan.

The curve shown in Fig. 2 contradicts the definition of topological dimension. Although it is a line (DT = 1), it has an intricate pattern that fills the entire space of the plane,

becoming indistinguishable of it (DT = 2). Considering the existence of such geometric

features, a new concept of dimension was necessary in order to describe them. Following work considered that the dimension of a feature S is given by the smallest possible union of n subsets that cover S, and each subset has a diameter equal to or smaller than δ and δ approaches 0 (Pruess, 1995). In practice, this abstract definition means that the size of a feature S is measured by covering it with regular geometric features with diagonal δ (usually squares) and the dimension of S is given by Eq. (1) (Pruess, 1995):

𝜇

= lim

𝛿→0

𝑛(𝛿)𝛿

𝐷

₍₁₎

Where n(δ) is the number of squares with diagonal δ necessary to cover the feature S whose dimension is to be determined, and µD is a constant of proportionality

(Pruess, 1995). Such dimension, firstly known as the Hausdorff-Besicovitch dimension (D), was later renamed as fractal dimension (Mandelbrot, 1983). Rearranging Eq. (1) and representing it in logarithmic form, we have the dimension D as in Eq. (2) (Pruess, 1995):

𝐷 = lim

log 𝑛(𝛿) − log 𝜇

log(1_𝛿)

(2)

In practice, it is not possible to reduce the size of the squares that cover the feature of interest to the limit 0. So, the above equation is reorganized and D is approximated by the slope of a line on a log-log graph as in Eq. (3):

171

log 𝑛(𝛿) = log 𝜇

− 𝐷 log 𝛿 (3)

To build a log n(δ) x log δ graph, appropriate values of δ are chosen, data points obtained are fitted with a straight line and its slope is used as an estimate of the Hausdorff- Besicovitch dimension (Fig. 3).

According to the concept of Hausdorff- Besicovitch dimension, a point has D = 0, a line D = 1 and a plane D = 2, coinciding with DT of these features. However, for complex

objects, D does not coincide with DT. For the Peano curve of Fig. 2, DT = 1 and D = 2; for the

Koch curve of Fig. 3, DT = 1 and D ≈ 1.26 (Mandelbrot, 1983). In both cases, the Hausdorff-

Besicovitch dimension is greater than the topological dimension of the curves (D > DT),

which is the definition of a fractal (a feature in which the Hausdorff-Besicovitch dimension is larger than the topological dimension). Note that D for the Koch curve is a fractional number (a ubiquitous characteristic in this type of feature); this is the reason why the term "fractal" was chosen by Mandelbrot.

Fig. 3: Procedure for estimating the Hausdorff-Besicovitch dimension of a Koch curve. (a) First five

iterations for making the curve. (b) The curve is covered by grids of squares with increasingly small diagonal δ. As δ decreases, the number of squares n(δ) needed to cover the curve increases. (c) The slope of line in the log n(δ) x log δ graph provides an estimate of D = 1.178. For a theoretical curve such as the one illustrated here, D can be determined exactly from the Hausdorff-Besicovitch equation (D ≈ 1.261; Mandelbrot 1983). Deviation from the theoretical value is inherent to the method applied, but can be reduced by using more grids.

References

Boots, B.N., Getis, A., 1988. Point Pattern Analysis, Scientific Geography 8. SAGE Publications Inc., Newbury Park, 92 pp.

Carranza, E.J.M., 2009. Controls on mineral deposit occurrence inferred from analysis of their spatial pattern and spatial association with geological features. Ore Geol. Rev. 35, 383– 400. doi:10.1016/j.oregeorev.2009.01.001

Diggle, P.J., 1983. Statistical Analysis of Spatial Point Patterns. Academic Press, London, 148 pp.

Fry, N., 1979. Random point distributions and strain measurement in rocks. Tectonophysics 60, 89–105. doi:10.1016/0040-1951(79)90135-5

Mandelbrot, B.B., 1983. The Fractal Geometry of Nature (Updated and Augmented Edition). Freeman, New York, 495 pp.

Pruess, S.A., 1995. Some Remarks on the Numerical Estimation of Fractal Dimension. In: Barton, C.C., La Pointe, P.R. (Eds.), Fractals in the Earth Sciences. Springer US, Boston, MA, pp. 65–75. doi:10.1007/978-1-4899-1397-5_3

173

APÊNDICE B – Material suplementar do CAPÍTULO 4

Appendix A. Supplementary figure

Fig. A1. Examples of texture classification as used in this study. (a) Disseminated alteration (photo

SQR-005-h_b1). (b) Sparse alteration (photo SQR-016-h_a1). (c) Infill (photo SSG-002-h_a1). Left: Reflected light. Right: bitmap of ore minerals (chalcopyrite for these samples). Scale bar: 200 μm.